Suppose we want to know if there exists a Belyi map whose branch pattern above , , is [1$39], [2$14], [7$4]. This means that should have root of order and roots of order , should have roots of order , should have poles of order , and should be unramified outside of {0,1,infinity}. We can determine if such exist (and if so, how many) as follows.
Found conjugacy class of -constellations (i.e. dessin), so there exists a Belyi map (unique up to equivalence) with branch pattern B.
Now let's check that d = [ g0, g1 ] has branch pattern B.
g0 indeed has cycle-structure [1,3$9] (a -cycle and -cycles)
g1 has cycle-structure [2$14] ( -cycles)
Has cycle structure [7$4] ( -cycles).
The Belyi map for d is indecomposable.
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Example with decompositions
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The Belyi map for S[1] has decompositions. With additional arguments, DecomposeDessin returns a list with information on each Fn, and a decomposition graph.
Decomposition graph:
F1 .. F5 have the same dessin so they represent the same Belyi map (of degree = ). The reason for listing all five is because their degree = 2 decomposition factors differ.
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